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We consider the 2D incompressible Euler equation on a bounded simply connected domain\Omega. We give sufficient conditions on the domain\Omegaso that for any initial vorticity\omega_{0} \in L^{\infty}(\Omega), the weak solutions are unique. Our sufficient condition is slightly more general than the condition that\Omegais aC^{1,\alpha}domain for some\alpha>0, with its boundary belonging toH^{3/2}(\mathbb{S}^{1}). As a corollary, we prove uniqueness forC^{1,\alpha}domains for\alpha >1/2and for convex domains which are alsoC^{1,\alpha}domains for some\alpha >0. Previously, uniqueness for general initial vorticity inL^{\infty}(\Omega)was only known forC^{1,1}domains with possibly a finite number of acute angled corners. The fundamental barrier to proving uniqueness below theC^{1,1}regularity is the fact that for less regular domains, the velocity near the boundary is no longer log-Lipschitz. We overcome this barrier by defining a new change of variable which we then use to define a novel energy functional.
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The quasi-periods of the Weierstrass zeta-function
We study the ratiop=\eta_{1}/\eta_{2}of the quasi-periods of the Weierstrass\zeta-function in dependence of the ratio\tau=\omega_{1}/\omega_{2}of the generators of the underlying rank-2 lattice. We will give an explicit geometric description of the map\tau\mapsto p(\tau). As a consequence, we obtain an explanation of a theorem by Heins who showed thatpattains every value in the Riemann sphere infinitely often. Our main result is implicit in the classical literature, but it seems not to be very well known.Essentially, this is an expository paper. We hope that it is easily accessible and may serve as an introduction to these classical themes.
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- Award ID(s):
- 2054987
- PAR ID:
- 10629570
- Publisher / Repository:
- EMS Press
- Date Published:
- Journal Name:
- L’Enseignement Mathématique
- Volume:
- 71
- Issue:
- 3
- ISSN:
- 0013-8584
- Page Range / eLocation ID:
- 335 to 364
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript
- Journal Article:
- https://doi.org/10.4171/LEM/1085
